Sharing a cake is not as easy as child’s play

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

09 July 2006

Sharing can be a very complicated process. Suppose you give a cake to two children (Alice and Beth) and ask them to share it fairly. The best way to go about it is to apply the “I cut, you choose principle”. That is one child (say Alice) cuts and then for the other (Beth) picks his piece first. This is called proportional, envy-free sharing.

It is “proportional” because Alice will make sure that the two pieces are as equal as possible (she knows if she doesn’t, Beth will pick the bigger piece) and “envy-free” because nobody goes away thinking that they were cheated. That is easy to do when there are only two people to share, but what happens when there are more?

In my school days one loaf of bread used to be shared amongst five people. We would alternate the person to cut each day and he would pick his piece last. This method was not perfect because the person who picked the first piece would select the largest one. But since the sharing was repeated every morning, the envy would resolve itself as we all tried to go for the first piece each day. But if it was a one-off sharing, a dispute would ensue and some people would walk away unsatisfied.

There is a fair mathematical method of sharing amongst any number of people without leaving any envy. However we shall not go into that now because cakes and bread are child’s play. You see, “we” adults deal with more important stuff – money!

Here is a scenario that is played out every day in Kenya. Three friends (Onyango, Kamau and Rop) go out for roast meat at the local market. They agree to share the cost of the meal equally. However, Onyango is asked to go to a nearby stall and buy some vegetable salad. He spends Sh40 on it. The meat is another Sh110. How much will Onyango pay at the butchery?

The total cost is Sh150 and this is to be divided by three, that is, each person must pay Sh50. But since Onyango had already paid Sh40 for the salad, he will only contribute only Sh10 for the meat. They can also decide to share the costs of the meat and the salad separately, but that would only complicate matters; 40 and 110 are not divisible by three!

The situation is a little more confusing if the three friends were to share the profit from a joint business project. Suppose they have been trading and withdrawing different amounts of money depending on personal needs. Onyango has taken out Sh50,000, Kamau Sh70,000 and Rop Sh90,000. At the end of the project, there is Sh300,000 in the bank. How would they share it?

The amounts withdrawn are part of the profit from the business project so they should be added to the Sh300,000 bank balance before sharing. Thus there would have been Sh510,000 if no withdrawals were made. Dividing this by three gives Sh170,000. Each person will thus get Sh170,000 minus the amount that he had taken out during the project. That is Onyango gets Sh120,000, Kamau goes away with Sh100,000 and Rop Sh80,000. That is easier than sharing a cake amongst three children.